\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 151, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2013/151\hfil Multiplicity of homoclinic solutions] {Multiplicity of homoclinic solutions for second-order Hamiltonian systems} \author[G. Bao, Z. Han, M. Yang \hfil EJDE-2013/151\hfilneg] {Gui Bao, Zhiqing Han, Minghai Yang} % in alphabetical order \address{Gui Bao \newline School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China} \email{baoguigui@163.com} \address{Zhiqing Han \newline School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China} \email{hanzhiq@dlut.edu.cn} \address{Minghai Yang \newline Department of Mathematics, Xinyang Normal University, Xinyang 464000, China} \email{ymh1g@126.com} \thanks{Submitted January 4, 2013. Published June 28, 2013.} \subjclass[2000]{37J45, 58E05, 34C37, 70H05} \keywords{Second order Hamilton system; Homoclinic solution; \hfill\break\indent variational method} \begin{abstract} By using a modified function technique and variational methods, we establish the existence of infinitely many homoclinic solutions for a second-order Hamiltonian system $\ddot{u}-L(t)u+F_u(t,u)=0$, for all $t\in \mathbb{R}$, where no coercive condition for $F(t,u)$ at infinity is imposed. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction and statement of main results} This article concerns the existence of homoclinic solutions for the following second-order Hamiltonian system $$\ddot{u}-L(t)u+F_u(t,u)=0, \quad \forall t\in \mathbb{R}, \label{e1.1}$$ where $u=(u_1,\dots,u_N)\in \mathbb{R}^N$, $L\in C(\mathbb{R},\mathbb{R}^{N^2})$ is a symmetric matrix-valued function and $F\in C^1(\mathbb{R}\times \mathbb{R}^N,\mathbb{R})$. Here, as usual, we say that a solution $u$ of system \eqref{e1.1} is a homoclinic solution (to 0) if $u\in C^2(\mathbb{R},\mathbb{R}^N)$, $u(t)\not\equiv0$, $u(t)\to 0$ and $\dot{u}(t)\to 0$ as $|t|\to \infty$. There have been many papers devoted to the homoclinic solutions of second order Hamiltonian systems via variational methods; see, e.g., \cite{Am,Ca,CR,DING,Ding1,IJ,OW,RA,RT,Tang,Yang1,Yang,Zhang,Yuan,ZOU} and the references therein. If $L$ and $F$ are $T$-periodic in $t$, Rabinowitz \cite{RA} obtains the existence of one homoclinic solution to system \eqref{e1.1} as a limit of $2kT$-periodic solutions. The methods and the results are extended by many further works; e.g. see \cite{CR} for a significant paper. If $L$ and $F$ are not periodic in $t$, the problem of existence of homoclinic solutions to system \eqref{e1.1} is quite different. We now recall some papers. In \cite{DING}, the author considers the case where $L(t)$ is not periodic and the corresponding linear part is not necessarily positive definite and proves that system \eqref{e1.1} possesses homoclinic solutions by extending the compact imbedding theorems in \cite{OW}. The case is also considered in \cite{Yang} but $F(t,u)$ is subquadratic satisfying a variant of the Ahmad-Lazer-Paul type condition. By using variant fountain theorem, the authors in \cite{Zhang} also investigate the case when $F(t,u)$ is subquadratic or superquadratic. We should point out that either in the superquadratic or the subquadratic case for $F(t,u)$, which is considered in the above mentioned papers, some kind of coercive conditions at infinity are needed. In this paper, by using variational methods, we obtain infinitely many homoclinic solutions of system \eqref{e1.1} without requiring any coercive condition or even any growth restriction for $F(t,u)$ at infinity when $F(t,u)$ is subquadratic. We introduce the following hypotheses. \begin{itemize} \item[(L1)] There exist $a>0$ and $r>0$ such that one of the following two conditions is true, \begin{itemize} \item[(i)] $L\in C^1(\mathbb{R},\mathbb{R}^N)$ and $|L'(t)|\leq a|L(t)|$ for all $|t|\geq r$, \item[(ii)] $L\in C^2(\mathbb{R},\mathbb{R}^N)$ and $L''(t)\leq aL(t)$ for all $|t|\geq r$, where $L'(t)=({\rm d}/{\rm d}t)L(t)$ and $L''(t)=({\rm d}^2/{\rm d}t^2)L(t)$. \end{itemize} \item[(L2)] There exists $\alpha<1$ such that $$l(t)|t|^{\alpha-2}\to \infty\text{as }|t|\to \infty,$$ where $l(t)$ is the smallest eigenvalue of $L(t)$; i.e., $$l(t):=\inf_{|\xi|=1,\,\xi\in \mathbb{R}^N }\langle L(t)\xi,\xi\rangle.$$ \item[(F1)] $F(t,u)\geq0$ for all $(t,u)\in \mathbb{ R}\times\mathbb{ R}^N$ and there exists a constant $1<\mu<2$ such that $$\langle F_u(t,u),u\rangle\leq \mu F(t,u),\quad \forall (t,u)\in\mathbb{R}\times\mathbb{R}^N.$$ \item[(F2)] $F(t,0)\equiv0$ and there exist constants $c_1>0, R_1>0$ and $\frac{1}{2}\leq v<1$ such that $$|F_u(t,u)|\leq c_1|u|^v, \quad \forall t \in\mathbb{R},\; |u|\leq R_1.$$ \item[(F3)] There exist constants $L_0>0,L_1>0,d_0>0$, where $L_1$ is sufficiently large (fixed below), such that $$F(t,u)\geq d_0|u|>0,\quad \forall t\in\mathbb{R},\; L_0\leq|u|\leq L_1.$$ \item[(F4)] $0<\underline{b}\equiv\inf_{t\in\mathbb{R},\, |u|=1}F(t,u)\leq\sup_{t\in\mathbb{R},\,|u|=1}F(t,u)\equiv \overline{b}<\infty$. \end{itemize} Here and in the sequel, $\langle\cdot,\cdot\rangle$ and $|\cdot|$ denote the standard inner product and the associated norm in $\mathbb{R}^N$ respectively. \begin{remark} \label{rmk1.1}\rm In fact, if we set $$M:=\tau_{\infty}\Big(4+(a_1^4+2a_2^4)\Big(L_0+\frac{8}{d_0(2-\mu)}\Big)^2 +8c_2\tau_{1+v}^{1+v}+8c_2\tau_{\mu}^{\mu}\Big)^{\frac{1}{2-s}}$$ then the constant $L_1$ in (F3) can be any constant bigger than $M$, where $s=\max\{1+v,\mu\}$, $\tau_{1+v}$, $\tau_{\mu}$ and $\tau_{\infty}$ are defined in Lemma \ref{sobolev}, $a_1, a_2$ are defined in the proof of Theorem \ref{thm1}, $c_2$ is defined in \eqref{e3.1}. \end{remark} Our main results are the following theorems. \begin{theorem} \label{thm1} Suppose that {\rm (L1)--(L2), (F1)--(F4)} are satisfied, and $F(t,u)$ is even in $u$. Then system \eqref{e1.1} has infinitely many homoclinic solutions. \end{theorem} \begin{theorem} \label{thm2} Suppose that $L(t)$ is positive for all $t$, and satisfies {\rm (L1)--(L2)}. Assume that $F(t,u)$ is even in $u$ and \begin{itemize} \item[(F5)] $\lim_{|u|\to 0}\frac{F(t,u)}{|u|^2}=\infty$ uniformly for $t\in\mathbb{R}$. \end{itemize} Then system \eqref{e1.1} has infinitely many homoclinic solutions which converge to zero. \end{theorem} \begin{remark} \label{rmk1.2}\rm We point out that there are natural functions $F(t,u)$ satisfying the conditions of Theorem \ref{thm1}. For example, $$F(t,u)=u^{6/5}e^{-\varepsilon u^2}.$$ It is easy to see that, for $\varepsilon>0$ small, $F(t,u)$ does not satisfy any of the coercive conditions for the problem \eqref{e1.1} in the above-mentioned papers (c.f. \cite{DING,Zhang,Yang}). \end{remark} \section{Variational settings and preliminaries}\label{section3} We first recall the variational settings for system \eqref{e1.1}. Denote by $\mathcal{A}$ the self-adjoint extension of the operator $-({\rm d}^2/{\rm d}t^2)+L(t)$ with the domain ${\mathcal{D}}({\mathcal{A}})\subset L^2:=L^2(\mathbb{R}, \mathbb{R}^N)$. Let $E:={\mathcal{D}}(|{\mathcal{A}}|^{1/2})$, the domain of $|{\mathcal{A}}|^{1/2}$, and define in $E$ the inner product and norm by $(u,v)_0:=(|{\mathcal{A}}|^{1/2}u, |{\mathcal{A}}|^{1/2}v)_2+(u,v)_2,\quad \|u\|_0:=(u,u)_0^{1/2},$ where, as usual, $(\cdot,\cdot)_2$ denotes the inner product of $L^2$. Then $E$ is a Hilbert space. The following lemma is proved in \cite{DING}. \begin{lemma}\label{sobolev} If $L(t)$ satisfies condition {\rm (L2)}, then $E$ is compactly embedded in $L^p:=L^p(\mathbb{R},\mathbb{R}^N)$ for $1\leq p\leq\infty$, which implies that there exists a constant $\tau_p> 0$ such that $$|u|_p\leq\tau_p\|u\|_0, ~\forall u\in E.$$ \end{lemma} By Lemma \ref{sobolev}, the spectrum $\sigma({\mathcal{A}})$ consists of only eigenvalues numbered in $\lambda_1\leq\lambda_2\leq\dots\to \infty$(counted in their multiplicities) and a corresponding system of eigenfunctions $\{e_n\}$, ${\mathcal{A}}e_n=\lambda_ne_n$, which forms an orthogonal basis of $L^2$. Assume that $\lambda_1,\ldots,\lambda_{n^-}<0$, $\lambda_{n^-+1}=\dots=\lambda_{\bar{n}}=0$, and let $E^-:=\operatorname{span}\{e_1,\ldots,e_{n^-}\}$, $E^0:=\operatorname{span}\{e_{n^-+1},\ldots,e_{\bar{n}}\}$ and $E^+:=\overline{\operatorname{span}\{e_{\bar{n}+1},\ldots\}}$. Then $E=E^-\oplus E^0\oplus E^+$. We introduce in $E$ the inner product $$(u,v):=(|{\mathcal{A}}|^{1/2}u, |{\mathcal{A}}|^{1/2}v)_2+(u^0,v^0)_2$$ and the norm $$\|u\|^2=(u,u)=\||{\mathcal{A}}|^{1/2}u\|_2^2+\|u^0\|_2^2,$$ where $u=u^-+u^0+u^+$ and $v=v^-+v^0+v^+\in E^-\oplus E^0\oplus E^+$. Then $\|\cdot\|$ and $\|\cdot\|_0$ are equivalent. From now on, the norm $\|\cdot\|$ in $E$ will be used. Hereafter, $(\cdot,\cdot)$ denotes the inner product in $E$ or the pairing between $E^*$ and $E$. Let $X$ be a Banach space with the norm $\|\cdot\|$ and $X=\overline{\oplus_{j\in N}X_j}$ with $\dim X_j<\infty$, for any $j\in \mathbb{N}$. Set $Y_k=\oplus_{j=1}^kX_j$ and $Z_k=\overline{\oplus_{j=k}^\infty X_j}$. Consider the following $C^1$-functional $\Phi_{\lambda}: X \to \mathbb{R}$ defined by $$\Phi_{\lambda}(u):=A(u)-\lambda B(u),~\lambda\in[1,2].$$ The following variant of the fountain theorem is established in \cite{ZOU}. \begin{proposition}\label{zou} Assume that the functional $\Phi_{\lambda}$ defined above satisfies the following conditions. \begin{itemize} \item[(T1)] $\Phi_{\lambda}$ maps bounded sets to bounded sets uniformly for $\lambda\in[1,2]$, $\Phi_\lambda(-u)=\Phi_\lambda(u)$ for all $(\lambda,u)\in [1,2]\times X$. \item[(T2)] $B(u)\geq0$ for all $u\in X$; $B(u)\to \infty$ as $\|u\|\to \infty$ in any finite dimensional subspace of $X$. \item[(T3)] There exist $\rho_k>r_k>0$ such that $$\alpha_k(\lambda):=\inf_{u\in Z_k,\|u\| =\rho_k}\Phi_\lambda(u)\geq0>\beta_k(\lambda) :=\max_{u\in Y_k,\|u\|=r_k}\Phi_\lambda(u),\quad \forall\lambda\in[1,2],$$ and $$\xi_k(\lambda):=\inf_{u\in Z_k,\|u\|\leq\rho_k}\Phi_\lambda(u)\to 0, \quad\text{as k\to \infty uniformly for \lambda\in [1,2]}.$$ \end{itemize} Then there exist $\lambda_n\to 1$, $u_{\lambda_n}\in Y_n$ such that $$\Phi_{\lambda_n}'|_{Y_n}(u_{\lambda_n})=0,\quad \Phi_{\lambda_n}(u_{\lambda_n})\to \eta_k\in[\xi_k(2),\beta_k(1)], \quad\text{as n\to \infty}$$ Particularly, if $\{u_{\lambda_n}\}$ has a convergent subsequence for every $k$, then $\Phi_1$ has infinitely many nontrivial critical points $\{u_k\}\subset X\backslash\{\theta\}$ satisfying $\Phi_1(u_k)\to 0^-$ as $k\to \infty$. \end{proposition} We shall use a result from \cite{RK}. For this purpose, we first recall the definition of genus. \begin{definition} \label{def2.1}\rm Let $X$ be a real Banach space and $A$ a subset of $X$. The set $A$ is said to be symmetric if $u\in A$ implies $-u\in A$. For a closed symmetric set $A$ which does not contain the origin, we define a genus $\gamma(A)$ of $A$ as the smallest integer $k$ such that there exists an odd continuous mapping from $A$ to $\mathbb{R}^k\setminus\{\theta\}$. If there does not exist such a $k$, we define $\gamma(A)=\infty$. Moreover, we set $\gamma(\emptyset) = 0$. Let $\Gamma_k$ denote the family of closed symmetric subsets $A$ of $X$ such that $0 \notin A$ and $\gamma(A)\geq k$. \end{definition} \begin{remark}[\cite{MW,RAB}] \label{rmk2.1} \rm 1. For any bounded symmetric neighborhood $\Omega$ of the origin in $\mathbb{R}^m$ it holds that $\gamma(\partial\Omega)=m$. 2. Let $A, B$ be closed symmetric subsets of $X$ which do not contain the origin. If there is an odd continuous mapping from $A$ to $B$, then $\gamma(A)\leq\gamma(B)$. \end{remark} The following proposition is established in \cite{RK}. \begin{proposition}\label{k} Let $X$ be an infinite dimensional Banach space and let $I\in C^1(X, \mathbb{R})$ satisfy the following two conditions: \begin{itemize} \item[(A1)] $I(u)$ is even, bounded from below, $I(\theta) = 0$ and $I(u)$ satisfies the Palais-Smale condition {\rm(PS)} \item[(A2)] For each $k \in\mathbb{N}$, there exists an $A_k\in \Gamma_k$ such that $\sup_{u\in A_k} I(u) < 0$. \end{itemize} Then $I(u)$ admits a sequence of critical points ${u_k}$ such that $I(u_k)\leq 0$, $u_k\neq\theta$ and $\lim_{k\to\infty} u_k = \theta$. \end{proposition} \section{Proofs of the main results } \subsection{Proof of Theorem \ref{thm1}} By (F1), (F2) and (F4), we obtain $$|F(t,u)|\leq c_2(|u|^{1+v}+|u|^{\mu}),\quad \forall (t,u)\in \mathbb{R}\times \mathbb{R}^N,\label{e3.1}$$ for some $c_2>0$. By (F3), there exists a constant $\delta_0>0$ such that $$F(t,u)\geq \frac{d_0}{2}|u|>0,\quad \forall t\in\mathbb{R},\; L_0\leq|u|\leq L_1+\delta_0.\label{e3.2}$$ Let $\chi\in C^\infty(\mathbb{R},\mathbb{R})$ such that $\chi(y)\equiv1$, if $y\leq L_1$, $\chi(y)\equiv0$, if $y\geq L_1+\delta_0$ and $\chi'(y)<0$, if $y\in(L_1,L_1+\delta_0)$. Set $$G(t,u):=\chi(|u|)F(t,u)+\frac{d_0}{2}(1-\chi(|u|))|u|.$$ Then $G\in C^1(\mathbb{R}\times\mathbb{R}^N,\mathbb{R})$ and $G(t,u)\geq0$ for all $(t,u)\in \mathbb{ R}\times\mathbb{ R}^N$. It is easily seen that $$\langle G_u(t,u),u\rangle=\chi(|u|)\langle F_u(t,u),u\rangle +\chi'(|u|)|u|(F(t,u)-\frac{d_0}{2}|u|)+\frac{d_0}{2}(1-\chi(|u|))|u|.$$ Hence,by (F1), \eqref{e3.2} and the definition of $\chi$, we have $$\langle G_u(t,u),u\rangle\leq \mu G(t,u),\quad \forall (t,u)\in \mathbb{R}\times \mathbb{R}^N.\label{e3.3}$$ Without loss of generality, we assume that $d_0\leq 1$. Combining \eqref{e3.1} and \eqref{e3.2}, we obtain $$G(t,u)\leq2c_2(|u|^{1+v}+|u|^{\mu}),\ \forall (t,u)\in \mathbb{R}\times \mathbb{R}^N,\label{e3.4}$$ and $$G(t,u)\geq\frac{d_0}{2}|u|>0,\quad \forall t\in\mathbb{R},\; |u|\geq L_0.\label{e3.5}$$ Let \begin{align*} \varphi(u) &= \frac{1}{2}\int_\mathbb{R}(|\dot{u}|^2+\langle L(t)u,u\rangle){\rm d}t -\int_\mathbb{R}G(t,u)\,{\rm d}t\\ &= \frac{1}{2}\|u^+\|-\frac{1}{2}\|u^-\|^2-\int_\mathbb{R}G(t,u)\,{\rm d}t\\ &= \varphi_1(u)+\varphi_2(u) \end{align*} where $\varphi_1(u)=\frac{1}{2}\|u^+\|-\frac{1}{2}\|u^-\|^2,~\varphi_2(u) =\int_\mathbb{R}G(t,u){\rm d}t$ for $u=u^-+u^0+u^+\in E$. By \cite{DING}, we have the following lemma. \begin{lemma}\label{kw} Suppose that {\rm (L1)--(L2), (F1)--(F4)} are satisfied. Then $\varphi_2\in C^1(E,\mathbb{R})$ and $\varphi_2':E\to E^*$ is compact. Moreover, \begin{gather*} (\varphi_2'(u),v) =\int_\mathbb{R}\langle G_u(t,u),v\rangle\,{\rm d}t,\\ (\varphi'(u),v)=(u^+,v^+)-(u^-,v^-)-\int_\mathbb{R}\langle G_u(t,u),v\rangle\,{\rm d}t \end{gather*} for all $u,v\in E=E^-\oplus E^0\oplus E^+$ with $u=u^-+u^0+u^+$ and $v=v^-+v^0+v^+$. Correspondingly, the nontrivial critical points of $\varphi$ in $E$ are the homoclinic solutions of the system $$\ddot{u}-L(t)u+G_u(t,u)=0,\quad \forall t\in \mathbb{R}.\label{e1.1*}$$ \end{lemma} To prove Theorem \ref{thm1} using Proposition \ref{zou}, we define the functionals \begin{gather} A(u)=\frac{1}{2}\|u^+\|^2,\quad B(u)=\frac{1}{2}\|u^-\|^2+\int_\mathbb{R}G(t,u)\ {\rm d}t,\label{e3.6} \\ \Phi_\lambda(u)=A(u)-\lambda B(u)=\frac{1}{2}\|u^+\|^2 -\lambda\Big(\frac{1}{2}\|u^-\|^2+\int_\mathbb{R}G(t,u)\ {\rm d}t\Big)\label{e3.7} \end{gather} for all $u=u^-+u^0+u^+\in E=E^-\oplus E^0\oplus E^+$ and $\lambda\in[1,2]$. By the similar arguments as in \cite{Zhang}, we obtain the following two Lemmas. For the completeness of this paper we will give their proofs. \begin{lemma}\label{wq} Suppose that {\rm (F1)--(F3)} are satisfied. Then $B(u)\geq0$ for all $u\in E$ and $B(u)\to \infty$ as $\|u\|\to \infty$ in any finite-dimensional subspace of $E$. \end{lemma} \begin{proof} By $G(t,u)\geq0$ and \eqref{e3.6}, we have $B(u)\geq0$. For any finite-dimensional subspace $E_0\subset E$, there exists a constant $\varepsilon>0$ such that $$m(\{t\in\mathbb{R}:|u(t)|\geq\varepsilon\|u\|\})\geq\varepsilon, \quad \forall u\in E_0\setminus\{\theta\},\label{e3.8}$$ where $m(\cdot)$ denotes the Lebesgue measure in $\mathbb{R}$. The proof of the claim is standard(e.g. see \cite{Zhang,Yang1}). Let $$\Lambda_u=\{t\in\mathbb{R}:|u(t)|\geq\varepsilon\|u\|\}, \quad \forall u\in E_0\setminus\{\theta\},$$ where $\varepsilon$ is given in \eqref{e3.8}. Then $$m(\Lambda_u)\geq\varepsilon,~~\forall u\in E_0\setminus\{\theta\}.\label{e3.9}$$ Combining with \eqref{e3.5} and \eqref{e3.9}, for any $u\in E_0$ with $\|u\|\geq L_0/\varepsilon$, we have \begin{align*} B(u)&= \frac{1}{2}\|u^-\|^2+\int_\mathbb{R}G(t,u)\ {\rm d}t\\ &\geq \int_{\Lambda_u}G(t,u)\ {\rm d}t\\ &\geq \int_{\Lambda_u}\frac{d_0}{2}|u|\ {\rm d}t\\ &\geq d_0\varepsilon\|u\|\cdot m(\Lambda_u)/2\\ &\geq d_0\varepsilon^2\|u\|/2. \end{align*} This implies that $B(u)\to \infty$ as $\|u\|\to \infty$ in any finite-dimensional subspace of $E_0\subset E$. The proof is completed. \end{proof} \begin{lemma} \label{jh} Suppose that {\rm (L2), (F1)-(F4)} are satisfied. Then there exist a positive integer $k_1$ and two sequences $00,\quad \forall k\geq k_1, \\ \xi_k(\lambda):=\inf_{u\in Z_k,\|u\|\leq\rho_k}\Phi_\lambda(u)\to 0 \quad\text{as$k\to \infty$uniformly for$\lambda\in[1,2]$}, \\ \beta_k(\lambda):=\max_{u\in Y_k,\|u\|=r_k}\Phi_\lambda(u)<0,\quad \forall k\in\mathbb{N}, \end{gather*} where$Y_k=\bigoplus_{j=1}^kX_j=\operatorname{span}\{e_1,\ldots,e_k\}$and$Z_k=\overline{\bigoplus_{j=k}^\infty X_j}=\overline{\operatorname{span}\{e_k,\ldots\}}$for all$k\in\mathbb{N}$. \end{lemma} \begin{proof} Let$l_k=\sup_{u\in Z_k,\|u\|=1}|u|_{1+v}^{1+v},\forall k\in \mathbb{N}$. Then$l_k\to 0$as$k\to \infty$(cf.\cite[Lemma 3.8]{WI}). Choose$k$large enough such that$Z_k\subset E^+$. Noticing (F2) and$F(t,u)=G(t,u)$as$|u|\leq R_1$, we have$G(t,u)\leq c_1|u|^{1+v}$for$|u|\leq R_1$. Therefore, for any$u\in Z_k$with$\|u\|\leq R_1/\tau_\infty$, we have $\Phi_\lambda(u) \geq \frac{1}{2}\|u\|^2-2\int_\mathbb{R}G(t,u)\ {\rm d}t \geq \frac{1}{2}\|u\|^2-2c_1l_k\|u\|^{v+1}.$ Set$\rho_k=(8c_1l_k)^{\frac{1}{1-v}}$. There exists a positive$k_1>\bar{n}+1$such that$\rho_k0. $$Noticing that \Phi_\lambda(\theta)=0, we have$$ 0\geq\inf_{u\in Z_k,\|u\|\leq\rho_k}\Phi_\lambda(u) \geq-2c_1l_k\rho_k^{v+1},\quad \forall k\geq k_1. $$Thus,$$ \xi_k(\lambda):=\inf_{u\in Z_k,\|u\|\leq\rho_k}\Phi_\lambda(u)\to 0 \quad \text{as $k\to \infty$ uniformly for }\lambda\in[1,2]. Since \dim Y_k<\infty, there exists a constant C_k>0 such that |u|_\mu\geq C_k\|u\|,~\forall u\in Y_k. By (F1) and (F4), for any k\in\mathbb{N} and |u|\leq1, we have G(t,u)\geq\underline{b}|u|^\mu. For any k\in\mathbb{N} and for all u\in Y_k with \|u\|<\tau_\infty^{-1}, we have \begin{align*} \Phi_\lambda(u) &\leq \frac{1}{2}\|u^+\|^2-\int_\mathbb{R}G(t,u)\ {\rm d}t\\ &\leq \frac{1}{2}\|u\|^2-\underline{b}|u|_\mu^\mu\\ &\leq \frac{1}{2}\|u\|^2-\underline{b}C_k^\mu\|u\|^\mu,\quad \forall\lambda\in[1,2]. \end{align*} Hence, for 00 such that |\xi_k(\lambda)|\leq1 for k\geq k_2. By \eqref{e3.10}, there exists n_0\in\mathbb{N} such that |\Phi_{\lambda_n}(u_n)|\leq2 for n\geq n_0 and k\geq \max\{k_1,~k_2\}. By (F1), (F3) and \eqref{e3.5}, we have \begin{align*} 2&\geq -\Phi_{\lambda_n}(u_n)\\&= \frac{1}{2}\Phi'_{\lambda_n}|_{Y_n}(u_n)u_n-\Phi_{\lambda_n}(u_n)\\ &\geq \lambda_n\int_{\Omega_n}\left[G(t,u_n)-\frac{1}{2}\langle G_u(t,u_n),u_n\rangle\right]\ {\rm d}t\\ &\geq \frac{\lambda_n(2-\mu)}{2}\int_{\Omega_n}G(t,u_n)\ {\rm d}t\\ &\geq \frac{d_0\lambda_n(2-\mu)}{4}\int_{\Omega_n}|u_n|\ {\rm d}t,\quad \forall n\in\mathbb{N}, \end{align*} where \Omega_n:=\{t\in\mathbb{R}:|u_n(t)|\geq L_0\}. Consequently, $$\int_{\Omega_n}|u_n|{~\rm d}t\leq \frac{8}{d_0(2-\mu)},\quad \forall n\in\mathbb{N},\; n\geq n_0.\label{e3.11}$$ For any n\in N, define \omega_n:\mathbb{R}\to \mathbb{R} by \omega_n=\begin{cases} 1,&t\in\Omega_n\\ 0,&t\notin\Omega_n. \end{cases} Noticing that \dim E^-\oplus E^0<\infty and \dim E^-<\infty, by the equivalence of the norms in finite-dimensional spaces, there exist two constants a_1, a_2>0 such that \begin{gather} |u_n^-+u_n^0|_1\leq a_1|u_n^-+u_n^0|_2,~|u_n^-+u_n^0|_\infty \leq a_1|u_n^-+u_n^0|_2,\label{e3.12} \\ \|u_n^-+u_n^0\|\leq a_1|u_n^-+u_n^0|_2,\label{e3.13} \\ |u_n^-|_1\leq a_2|u_n^-|_2 ,~|u_n^-|_\infty \leq a_2|u_n^-|_2 ,\label{e3.14} \\ \|u_n^-\|\leq a_2|u_n^-|_2.\label{e3.15} \end{gather} By Lemma \ref{sobolev}, \eqref{e3.11} and the H\"older inequality, we have \begin{align*} |u_n^-+u_n^0|_2^2 &= (u_n^-+u_n^0,u_n)_2\\ &= (u_n^-+u_n^0,(1-\omega_n)u_n)_2+(u_n^-+u_n^0,\omega_nu_n)_2\\ &\leq |(1-\omega_n)u_n|_\infty|u_n^-+u_n^0|_1+|\omega_nu_n|_1|u_n^-+u_n^0|_\infty\\ &\leq a_1\Big(L_0+\frac{8}{d_0(2-\mu)}\Big)|u_n^-+u_n^0|_2,\quad \forall n\in\mathbb{N},\; n\geq n_0. \end{align*} By \eqref{e3.13}, we obtain that $$\|u_n^-+u_n^0\|\leq a_1^2\Big(L_0+\frac{8}{d_0(2-\mu)}\Big),\quad \forall n\in\mathbb{N}.\label{e3.16}$$ Similarly, by Lemma \ref{sobolev}, \eqref{e3.14} \eqref{e3.15} and the H\"older inequality, we have $$\|u_n^-\|\leq a_2^2\Big(L_0+\frac{8}{d_0(2-\mu)}\Big),\quad \forall n\in\mathbb{N},\;n\geq n_0.\label{e3.17}$$ Without loss of generality, we assume that \|u_n\|\geq1. Then by Lemma \ref{sobolev}, \eqref{e3.4} \eqref{e3.16} and \eqref{e3.17}, for all n \in\mathbb{N}, n\geq n_0, we obtain \begin{align*} \|u_n\|^2 &= \|u_n^+\|^2+\|u_n^-+u_n^0\|^2\\ &= 2\Phi_{\lambda_n}(u_n)+\lambda_n\|u_n^-\|^2 +\|u_n^-+u_n^0\|^2+2\lambda_n\int_{\mathbb{R}}G(t,u_n)\ {\rm d}t\\ &\leq 4+(a_1^4+2a_2^4)\Big(L_0+\frac{8}{d_0(2-\mu)}\Big)^2 +8c_2(\tau_{1+v}^{1+v}\|u_n\|^{1+v}+\tau_{\mu}^{\mu}\|u_n\|^{\mu})\\ &\leq \Big(4+(a_1^4+2a_2^4)\Big(L_0+\frac{8}{d_0(2-\mu)}\Big)^2 +8c_2\tau_{1+v}^{1+v}+8c_2\tau_{\mu}^{\mu}\Big)\|u_n\|^{s}, \end{align*} where s=\max\{1+v,\mu\}. By noting that 1<\mu<2 and \frac{1}{2}\leq v<1, we have $$\|u_n\|\leq \Big(4+(a_1^4+2a_2^4)\Big(L_0+\frac{8}{d_0(2-\mu)}\Big)^2 +8c_2\tau_{1+v}^{1+v}+8c_2\tau_{\mu}^{\mu}\Big)^{\frac{1}{2-s}},\label{e3.18}$$ where the constant does not depend on L_1. Since E is embedded compactly into L^p for 1\leq p\leq\infty, by a standard argument, we obtain that \{u_n\}_{n=1}^{\infty} possesses a strong convergent subsequence in E for each k\geq\max\{ k_1,k_2\}. Hence, by Proposition \ref{zou}, system \eqref{e1.1*} possesses infinitely many homoclinic solutions. By Lemma \ref{jh} and \eqref{e3.10}, we know that \Phi_{\lambda_n}(u_{\lambda_n}^k) is bounded uniformly for \forall k\geq\max\{ k_1,k_2\}. Set M:=\tau_{\infty}\Big(4+(a_1^4+2a_2^4)\Big(L_0+\frac{8}{d_0(2-\mu)}\Big)^2 +8c_2\tau_{1+v}^{1+v}+8c_2\tau_{\mu}^{\mu}\Big)^{\frac{1}{2-s}}. $$By \eqref{e3.18} we obtain \|u^k\|\leq M,~\forall k\geq \max\{k_1, k_2\}, where u^k is the limit of \{u_n^k\}_{n=1}^{\infty}. Therefore, there exists a constant M>0 independent of L_1 such that |u^k|_\infty\leq M,~\forall k\geq \max\{k_1, k_2\}. Combining this with F(t,u)=G(t,u) for |u|\leq L_1, we know that system \eqref{e1.1} possesses infinitely many homoclinic solutions if L_1\geq M. The proof is complete. \end{proof} \begin{proof}[Proof of Theorem \ref{thm2}] Let M_0>0, and let \chi\in C^\infty(\mathbb{R},\mathbb{R}) and C>0 be such that \chi(y)\equiv1, if y\leq M_0; \chi(y)\equiv0, if y\geq M_0+1; and |\chi'(y)|0. Let$$ \widetilde{\varphi}(u)=\frac{1}{2}\int_{\mathbb{R}}(|\dot{u}|^2+\langle L(t)u,u\rangle){\rm d}t-\int_{\mathbb{R}}G(t,u)\,{\rm d}t. Then \widetilde{\varphi}\in C^1(E,\mathbb{R}) and the nontrivial critical points of \widetilde{\varphi} in E are the homoclinic solutions of system $$\ddot{u}-L(t)u+G_u(t,u)=0,\quad \forall t\in \mathbb{R}.\label{e1.1**}$$ Let \begin{align*} {\psi}(u) &= \frac{1}{2}\int_{\mathbb{R}}(|\dot{u}|^2+\langle L(t)u,u\rangle){\rm d}t-\chi(|u|)\int_{\mathbb{R}}G(t,u)\,{\rm d}t\\ &= \frac{1}{2}\|u\|^2-\chi(|u|)\int_{\mathbb{R}}G(t,u)\,{\rm d}t. \end{align*} Then, {\psi}\in C^1(E,\mathbb{R}). For \|u\|\geq \tau^{-1}_\infty (M_0+1), we have {\psi}(u)=\frac{1}{2}\|u\|^2, which implies that {\psi}(u)\to\infty as \|u\|\to\infty. Hence {\psi} is coercive on E. Then {\psi}(u) is bounded from below and, by noticing Lemma \ref{sobolev}, it satisfies the (PS) condition. By \eqref{e3.19}, it is easy to see that {\psi}(u) is even and {\psi}(\theta)=0. This shows that ({\rm A}_1) holds. By (F4), for any \varepsilon>0, there exists \delta>0, such that F(t,u)\geq\varepsilon^{-1}|u|^2, |u|\leq\delta. For any given k, let E_k:=\operatorname{span}\{e_1,\ldots,e_k\}. Then there exists a constant \eta_k such that |u|_2\geq\eta_k\|u\| for u\in E_k. Therefore, for any u\in E_k with \|u\|=\rho<\min\{\tau^{-1}_{\infty}M_0,\tau^{-1}_{\infty}\delta, 2\varepsilon^{-1}\eta_k\},  where $\varepsilon$ is small enough, we have \begin{align*} {\psi}(u)&= \frac{1}{2}\|u\|^2-\chi(|u|)\int_{\mathbb{R}}G(t,u)\,{\rm d}t\\ &\leq \frac{1}{2}\|u\|^2-\varepsilon^{-1}\eta_k^2\|u\|^2 < 0. \end{align*} Then $A:=\{u\in E_k:\|u\|=\rho\}\subset\{u\in X:\psi(u)<0\}$. By Remark \ref{rmk2.1}, we have that $\gamma(A)=k$ and $\gamma(\{u\in X:\psi(u)<0\})\geq\gamma(A)=k$. Setting $A_k=\{u\in X:\psi(u)<0\}$, then $A_k\in\Gamma_k$ and $\sup_{u\in\Gamma_k}\psi(u)<0$. This shows that $({\rm A}_2)$ holds. Hence, by Proposition \ref{k}, we obtain that $\psi$ admits a sequence of nontrivial solutions $\{u_k\}$ such that $\lim_{k\to\infty} u_k = \theta$. Then there exists $k_1>0$ such that $\|u_k\|\leq\tau^{-1}_{\infty}M_0$ for $k\geq k_1$. Since $\widetilde{\varphi}=\psi$ for $|u|\leq M_0$, we know that $\widetilde{\varphi}$ possesses infinitely many nontrivial nontrivial critical points $\{u_k\}$ for $k\geq k_1$. Therefore, \eqref{e1.1**} possesses infinitely many nontrivial solutions. That is, system \eqref{e1.1} has infinitely many solutions by noting that $F(t,u)=G(t,u)$ for $|u|\leq M_0$. 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